Resolvent formalism

In mathematics and theoretical physics, the resolvent (sometimes also called Green operator ) is the inverse of a shifted by a complex number z linear operator or a matrix. The amount of the Z values ​​for which the inverse is well defined, the resolvent of the operator; the complement of this set is its spectrum. Applications relate to all aspects of operator theory in functional analysis, in particular the perturbation theory.

Definition

For a linear operator (or matrix) we define the resolvent as the complement of the spectrum of, ie as the set of complex numbers, for which the operator is limited to invertible. The resolvent set is open as a complement of the spectrum. On the resolvent set is defined by the resolvent

Many authors use as a definition of the resolvent, which merely inverts the sign.

Properties and Applications

The resolvent is an operator valued analytic function and can, whereby the spectral radius is represented by the Neumann series:

The resolvent is, inter alia, used to describe eigenvalue developments of interfered operators, for example, the developments of Rellich - Kato and Strutt Schrödinger.

Resolventenidentitäten

Helpful in calculations are the first and second Resolventenidentität. It follows from the first by inversion Resolventenidentität

And follows by inversion of the second Resolventenidentität